What is the distributive property in 3rd, 4th, and 5th grade math?

The distributive property says you can multiply a sum by breaking it into pieces, multiplying each piece, and adding the results. In symbols: a × (b + c) = (a × b) + (a × c). In a picture, it's a rectangle chopped into two smaller rectangles — the total area equals the sum of the two piece areas. In 3rd–5th grade, this is the single most useful idea for turning a hard multiplication into two easy ones.

What are partial products?

Partial products are the smaller multiplications you get when you split a bigger multiplication using place value. For example, 23 × 6 becomes (20 × 6) + (3 × 6) = 120 + 18 = 138. The 120 and 18 are the two partial products. Every partial product is a rectangle in the area model.

What is the box method in multiplication?

The box method is the area-model rectangle written as a table. Break each factor into place-value pieces along the sides of a grid, multiply each cell, then add all the cells together. For 23 × 47 you get a 2×2 grid with the pieces 20×40, 20×7, 3×40, 3×7 — add all four to get 1,081. The box method is the same math as the standard algorithm, just laid out where every step is visible.

How does the distributive property connect to the standard multiplication algorithm?

The standard multiplication algorithm — the one with the carries you learn in 4th and 5th grade — is a compressed form of partial products. Every line you write in the algorithm is one row of partial products. Understanding the distributive property first (with the box method) makes the standard algorithm click, because students see what each carry actually means: it is the tens digit of a partial product being combined into the next column.

When do children first learn the distributive property?

In the Common Core standards used across most US schools, the distributive property is introduced in 3rd grade (splitting a rectangle) and used heavily in 4th and 5th grade to multiply multi-digit numbers. It shows up again in 6th–8th grade for expressions like 3(x + 4), and again in high school algebra for FOIL and polynomial multiplication. The same picture — a rectangle split into pieces — is used every single time.

Where can I find a small-group math class in NYC that teaches this way?

SOMATH runs the Kid Einsteins program (grades 3–5) at 226 W 79th St on the Upper West Side of NYC. Classes are small groups of 6–8 students and use area models, the box method, and partial products as the core mental picture for every multiplication. Call (646) 668-6151 or email hello@schoolofmath.us to book a free evaluation.

Kid Einsteins · Distributive Property · Grades 3–5 · NYC Math Class

Distributive Thinking & Partial Products — 25 Practice Questions with Theory & Hidden Answers (Kid Einsteins)

The single most useful idea in grade-school multiplication: split a hard problem into two easier ones. This class-ready lesson teaches the distributive property and partial products using area models and the box method, then finishes with 25 practice questions — 10 easy, 10 medium, 5 hard — each with click-to-reveal step-by-step answers. Built for the SOMATH Kid Einsteins program (grades 3–5) on the Upper West Side of Manhattan.

2026-07-24 · By the SOMATH team · 226 W 79th St, UWS · (646) 668-6151

This is Class 4 of the SOMATH Kid Einsteins multiplication arc. In Class 3 we drilled the three trickiest times tables (×7, ×8, ×9) using the area model. In this class we take the same picture — a rectangle chopped in two — and turn it into the engine for every hard multiplication your student will ever meet: multi-digit numbers, algebraic expressions, even fractions.

The main idea is only one sentence: if a multiplication is too hard to see in one step, split one of the numbers into two friendlier pieces, multiply each piece, and add. That’s distributive thinking. The two smaller multiplications you get are called partial products. This lesson makes both ideas concrete, then drills them with 25 questions. Written by the same team that teaches Kid Einsteins at SOMATH, run by cofounders Marcelo Ambrozio (Northwestern) and Vivianne Wright (Harvard).

How to use this in a class (40–50 min):

Warm-up: read the theory aloud and draw one split rectangle together (10 min).
Work through the box-method example 23 × 47 as a class (10 min).
Students attempt the 25 questions with the answers hidden (20 min).
Reveal answers together and re-teach any that missed 3+ students (10 min).

What’s in this lesson

The one-sentence big idea
Theory: what the distributive property really says
Theory: partial products (splitting by place value)
The box method — the area model as a table
Two-digit × two-digit with the box method
How this becomes the standard algorithm
Six friendly splits every student should know
25 practice questions (10 easy, 10 medium, 5 hard)
Answer key summary
About SOMATH & Kid Einsteins

1. The one-sentence big idea

Distributive thinking: If a multiplication is hard, split one number into two friendly pieces, multiply each piece separately, then add the two answers.

Every technique in this class — area models, partial products, the box method, and the standard algorithm — is the same idea in different clothing. Once a student sees that a multiplication is a rectangle that can be chopped in two, hard multiplication becomes almost automatic.

2. Theory: what the distributive property really says

Split the sum, then add the pieces

The distributive property says that multiplying a number by a sum is the same as multiplying by each piece of the sum and adding the results:

a × (b + c) = (a × b) + (a × c)

In everyday language: “multiply, then add” equals “add, then multiply.” As a picture, it’s a rectangle sliced into two smaller rectangles.

Example: 6 × 13. The number 13 is not on any times table you memorized — but 13 = 10 + 3, and both of those are easy:

6 × 13 = 6 × (10 + 3) = (6 × 10) + (6 × 3) = 60 + 18 = 78

×	10	3
6	60	18

Split 13 into 10 + 3. Green box: 6 × 10 = 60. Gold box: 6 × 3 = 18. Add: 60 + 18 = 78.

The magic is not the property — it’s the freedom. You get to pick the split. Always pick the split that lands you on numbers you already know.

3. Theory: partial products (splitting by place value)

The place-value split is the one to memorize

A partial product is one of the smaller multiplications you get after splitting. The most useful split is always by place value: break a two-digit number into its tens and its ones.

Example: 27 × 4. Split 27 into 20 + 7:

27 × 4 = (20 × 4) + (7 × 4) = 80 + 28 = 108

×	20	7
4	80	28

Two partial products: 80 and 28. Add them to get 108.

Why place value? Because multiplying by 10, 20, 30, 100, 200, and so on is just a smaller multiplication followed by adding zeros. So splitting by tens and ones guarantees that at least one of your partial products is easy.

20 × 4 = 2 × 4 × 10 = 80
30 × 6 = 3 × 6 × 10 = 180
40 × 50 = 4 × 5 × 100 = 2000

4. The box method — the area model as a table

Rows, columns, and one cell per partial product

The box method is the area-model rectangle written as a grid. Break the factors into place-value pieces along the top and left of a table. Fill in each cell with that row’s piece times that column’s piece. Add all the cells.

Example: 46 × 7. Split 46 into 40 + 6. One row, two columns:

×	40	6
7	280	42

Two partial products: 7 × 40 = 280 and 7 × 6 = 42. Total: 280 + 42 = 322.

The box method has two virtues: it never hides a step, and it works with any size number. You can split into three pieces, four pieces, hundreds and tens and ones. Same picture every time.

5. Two-digit × two-digit with the box method

Two-by-two grid, four partial products

When both factors are two-digit numbers, split both of them. You get a two-by-two box with four partial products.

Example: 23 × 47. Split: 23 = 20 + 3 and 47 = 40 + 7.

×	40	7
20	800	140
3	120	21

Four partial products, one per cell. Add all four to get the answer.

23 × 47 = (20 × 40) + (20 × 7) + (3 × 40) + (3 × 7) = 800 + 140 + 120 + 21 = 1,081

Notice: every partial product is a fact you already know. 2 × 4 = 8 makes 20 × 40 = 800. 3 × 7 = 21 stays 21. The box method is nothing more than four small multiplications plus one addition.

6. How this becomes the standard algorithm

The stacked algorithm is compressed partial products

When you learn to stack numbers and “carry the 1” in 4th or 5th grade, you are doing the same partial products — just compressed. Every line of the stacked algorithm is a partial product.

Example: 23 × 47 the stacked way (this reads bottom-up, right-to-left):

23 × 47 ______ 161 ← 23 × 7 = (20 × 7) + (3 × 7) = 140 + 21 = 161 + 920 ← 23 × 40 = (20 × 40) + (3 × 40) = 800 + 120 = 920 ______ 1,081

Two lines instead of four cells — but the same four partial products are living inside those two lines. The box method makes them visible; the algorithm hides them for speed. Both are correct. Teach the box first, then the algorithm makes sense.

7. Six friendly splits every student should know

Pick the split so the pieces are easy

Times 11: Split into 10 + 1. 7 × 11 = 70 + 7 = 77.
Times 12: Split into 10 + 2. 6 × 12 = 60 + 12 = 72.
Times 15: Split into 10 + 5. 8 × 15 = 80 + 40 = 120.
Times 19: Split into 20 − 1 (subtract instead of add). 6 × 19 = 120 − 6 = 114.
Times 25: Split into 20 + 5 or think “a quarter of 100.” 8 × 25 = 160 + 40 = 200.
Times 99: Split into 100 − 1. 7 × 99 = 700 − 7 = 693.

The general rule: split so that one piece is a multiple of 10 (or 100). Multiplying by 10 or 100 is free — the rest is just an easier fact.

Class summary in one sentence: Split one number into two friendly pieces — usually tens and ones — multiply each piece, then add.

25 Practice Questions

10 easy, 10 medium, 5 hard. Every question can be solved with a split of your choosing — try to pick the friendliest split each time. Click Show answer & solution to check your reasoning.

Question 1 Easy

Use the split 14 = 10 + 4 to compute 3 × 14.

Answer: 42

3 × 14 = 3 × (10 + 4) = (3 × 10) + (3 × 4) = 30 + 12 = 42

Two partial products: 30 and 12. Splitting by place value turns a “3 × 14” step into two facts your student already knows.

Question 2 Easy

Use the split 16 = 10 + 6 to compute 5 × 16.

Answer: 80

5 × 16 = (5 × 10) + (5 × 6) = 50 + 30 = 80

Partial products 50 and 30. Notice both are multiples of ten — sometimes the split gives you two very tidy pieces.

Question 3 Easy

Ms. Chen buys 6 packs of markers. Each pack has 12 markers. How many markers does she have in total?

Answer: 72 markers

6 × 12 = (6 × 10) + (6 × 2) = 60 + 12 = 72

Split 12 into 10 + 2. Two partial products (60 and 12), quick addition, done.

Question 4 Easy

A minivan holds 7 people. A tour company runs 13 minivans. How many people can ride in total?

Answer: 91 people

7 × 13 = (7 × 10) + (7 × 3) = 70 + 21 = 91

Split 13 into 10 + 3. The two partial products are 70 and 21; add them to get 91.

Question 5 Easy

Use place-value splitting to compute 8 × 15. Show the two partial products.

Answer: 120

8 × 15 = (8 × 10) + (8 × 5) = 80 + 40 = 120

Partial products 80 and 40. Both are multiples of 10 — that’s the friendly split of 15 doing its job.

Question 6 Easy

A rectangular garden is 4 feet wide and 18 feet long. What is its area?

Answer: 72 square feet

4 × 18 = (4 × 10) + (4 × 8) = 40 + 32 = 72

Area is length × width. Split 18 into 10 + 8. The two partial products (40 and 32) are literally the two smaller rectangles the garden splits into.

Question 7 Easy

Use the split 19 = 20 − 1 to compute 6 × 19.

Answer: 114

6 × 19 = (6 × 20) − (6 × 1) = 120 − 6 = 114

Splits can subtract too. Because 19 is one less than 20, it’s often easier to overshoot to 20 and take one copy back.

Question 8 Easy

A pack of stickers has 25 stickers. Ana buys 4 packs. How many stickers does she have?

Answer: 100 stickers

4 × 25 = (4 × 20) + (4 × 5) = 80 + 20 = 100

Split 25 into 20 + 5. Partial products 80 and 20. Nice round total — that’s the ×25 pattern (4 groups of 25 = 100 = a full dollar).

Question 9 Easy

A carton holds 6 eggs. A store receives 17 cartons. How many eggs is that in total?

Answer: 102 eggs

6 × 17 = (6 × 10) + (6 × 7) = 60 + 42 = 102

Split 17 into 10 + 7. Both partial products (60 and 42) are facts your student already knows from earlier Kid Einsteins classes.

Question 10 Easy

Draw the box method for 5 × 24. What are the two partial products, and what is the answer?

Answer: partial products 100 and 20; total 120

×	20	4
5	100	20

5 × 24 = (5 × 20) + (5 × 4) = 100 + 20 = 120

Two cells, two partial products. This box is the same picture as a 5-by-24 rectangle cut into a 5-by-20 piece and a 5-by-4 piece.

Question 11 Medium

Compute 23 × 6 using partial products.

Answer: 138

23 × 6 = (20 × 6) + (3 × 6) = 120 + 18 = 138

Two partial products: 120 and 18. This is exactly what a 5th grader is doing in their head when they “multiply, carry the 1.”

Question 12 Medium

A school orders 36 pizzas, and each pizza is cut into 8 slices. How many slices are there in total?

Answer: 288 slices

8 × 36 = (8 × 30) + (8 × 6) = 240 + 48 = 288

Split 36 into 30 + 6. Partial products 240 and 48.

Question 13 Medium

Marcus reads 27 pages per day for 7 days. How many pages does he read in the week?

Answer: 189 pages

7 × 27 = (7 × 20) + (7 × 7) = 140 + 49 = 189

Split 27 into 20 + 7. Notice that 7 × 7 = 49 is one of the trickier facts from Class 3 — that’s why we drilled it.

Question 14 Medium

A theater has 14 rows and 15 seats in each row. How many seats does the theater have?

Answer: 210 seats

×	10	5
10	100	50
4	40	20

14 × 15 = (10 × 10) + (10 × 5) + (4 × 10) + (4 × 5) = 100 + 50 + 40 + 20 = 210

Two-digit × two-digit means four partial products. This is our first taste of the 2×2 box — the same picture we’ll use for 23 × 47.

Question 15 Medium

Compute 45 × 12 using the box method.

Answer: 540

×	40	5
10	400	50
2	80	10

45 × 12 = 400 + 50 + 80 + 10 = 540

Four partial products: 400, 50, 80, 10. Notice how quickly they add when you group the round ones first: 400 + 80 = 480, then 50 + 10 = 60, then 480 + 60 = 540.

Question 16 Medium

A gym class has 25 students in each of 8 sections. How many students are in gym class in total?

Answer: 200 students

8 × 25 = (8 × 20) + (8 × 5) = 160 + 40 = 200

Split 25 into 20 + 5. Alternate shortcut: 4 groups of 25 make 100, so 8 groups make 200. Both routes give the same answer — that’s the point of distributive thinking.

Question 17 Medium

Use the split 99 = 100 − 1 to compute 7 × 99.

Answer: 693

7 × 99 = (7 × 100) − (7 × 1) = 700 − 7 = 693

Multiplying by 99 looks scary but is really “multiply by 100, then subtract one copy.” The distributive property works with subtraction too: a × (b − c) = (a × b) − (a × c).

Question 18 Medium

A parking garage has 18 floors, and each floor holds 24 cars. How many cars can the garage hold?

Answer: 432 cars

×	20	4
10	200	40
8	160	32

18 × 24 = 200 + 40 + 160 + 32 = 432

Four partial products: 200, 40, 160, 32. Grouped adds: 200 + 160 = 360, then 40 + 32 = 72, then 360 + 72 = 432.

Question 19 Medium

Which is bigger, 6 × 47 or 7 × 39? Use partial products to compare.

Answer: 6 × 47 is bigger (282 vs. 273, a difference of 9)

6 × 47 = (6 × 40) + (6 × 7) = 240 + 42 = 282 7 × 39 = (7 × 40) − (7 × 1) ← split 39 as 40 − 1 = 280 − 7 = 273

Two different products, both handled by splitting into a friendly form. Then compare: 282 > 273 by 9.

Question 20 Medium

A soccer league has 16 teams. Each team has 23 players. How many players are in the league?

Answer: 368 players

×	20	3
10	200	30
6	120	18

16 × 23 = 200 + 30 + 120 + 18 = 368

Four partial products, all small. Grouped adds: 200 + 120 = 320, then 30 + 18 = 48, then 320 + 48 = 368.

Question 21 Hard

Compute 23 × 47 using the box method. Show the four partial products.

Answer: 1,081

×	40	7
20	800	140
3	120	21

23 × 47 = (20 × 40) + (20 × 7) + (3 × 40) + (3 × 7) = 800 + 140 + 120 + 21 = 1,081

Four partial products: 800, 140, 120, 21. Add them any order you like. This is the “canonical” two-digit × two-digit example — every student should be able to write this grid from memory by the end of Class 4.

Question 22 Hard

A school buys 36 boxes of notebooks. Each box holds 48 notebooks. How many notebooks did the school buy?

Answer: 1,728 notebooks

×	40	8
30	1,200	240
6	240	48

36 × 48 = 1,200 + 240 + 240 + 48 = 1,728

Four partial products, two of which happen to be equal (240 appears twice). Grouped adds: 1,200 + 240 + 240 = 1,680, then 1,680 + 48 = 1,728.

Question 23 Hard

Fill in the missing partial product so that 28 × 15 = 200 + \_\_\_ + 80 + 40. What is the missing partial product, and what is the total?

Answer: missing partial product = 100; total = 420

28 × 15 = (20 × 10) + (20 × 5) + (8 × 10) + (8 × 5) = 200 + 100 + 80 + 40 = 420

×	10	5
20	200	100
8	80	40

The missing cell is 20 × 5 = 100. This kind of “find the missing partial product” problem is the fastest way to check that a student really understands the box method — not just how to fill it in, but how to read it.

Question 24 Hard

A rectangular auditorium is being tiled. The floor is 34 feet by 26 feet. Each tile is exactly 1 square foot. How many tiles are needed? Use the box method and list your four partial products.

Answer: 884 tiles

×	20	6
30	600	180
4	80	24

34 × 26 = 600 + 180 + 80 + 24 = 884

Area of the auditorium in square feet is the same as the number of 1-by-1 tiles. Four partial products: 600, 180, 80, 24. The box method is literally how a contractor would break up the floor to tile it in sections.

Question 25 Hard

Challenge: A gardener has a rectangular plot that is 25 feet by 18 feet. She adds a 2-foot-wide walking path along the entire 25-foot side, making the plot 25 feet by 20 feet. What is the area of the new plot, and what is the area of just the path? Use distributive thinking.

Answer: new plot area = 500 sq ft; path area = 50 sq ft

Original plot: 25 × 18 = (25 × 10) + (25 × 8) = 250 + 200 = 450 sq ft New plot: 25 × 20 = 500 sq ft (a friendly multiple of 100) Path area = new − original = 500 − 450 = 50 sq ft Or directly: 25 × 2 = 50 sq ft ← the path IS a 25-by-2 rectangle

Two ways to see the same answer. Distributively: 25 × 20 = 25 × (18 + 2) = (25 × 18) + (25 × 2) = 450 + 50 = 500. That last line is the whole idea of this lesson in one equation: adding a piece to a rectangle’s side is exactly the same as adding a new rectangle of partial products. This is also the picture your student will use again when they FOIL (x + 2)(x + 5) in Algebra 1 — same rectangle, same split, same answer.

Answer key summary

Q#	Answer	Q#	Answer	Q#	Answer
1	42	10	120	19	6×47 (282)
2	80	11	138	20	368
3	72	12	288	21	1,081
4	91	13	189	22	1,728
5	120	14	210	23	100; total 420
6	72	15	540	24	884
7	114	16	200	25	500 & 50
8	100	17	693
9	102	18	432

About SOMATH & Kid Einsteins

SOMATH — School of Math is a math-focused school on the Upper West Side of Manhattan for students in grades 1–12. The Kid Einsteins program (grades 3–5) drills multiplication and division fluency, fractions and decimals, area and perimeter, ratios, and structured word problems in small groups of 6–8 students. This is Class 4 of the Kid Einsteins arc — the natural sequel to Class 3 (×7, ×8, ×9 with area models).

Classes are taught by cofounder Marcelo Ambrozio (Northwestern-trained, 15+ years teaching math in NYC) and the SOMATH team.

Location: 226 W 79th St, 1st Floor, New York, NY 10024 (Upper West Side)
Phone: (646) 668-6151
Email: hello@schoolofmath.us

Book a free evaluation for your child — we’ll assess where they are and place them in the right small group.

Distributive Thinking & Partial Products — 25 Practice Questions with Theory & Hidden Answers (Kid Einsteins)

What’s in this lesson

1. The one-sentence big idea

2. Theory: what the distributive property really says

Split the sum, then add the pieces

3. Theory: partial products (splitting by place value)

The place-value split is the one to memorize

4. The box method — the area model as a table

Rows, columns, and one cell per partial product

5. Two-digit × two-digit with the box method

Two-by-two grid, four partial products

6. How this becomes the standard algorithm

The stacked algorithm is compressed partial products

7. Six friendly splits every student should know

Pick the split so the pieces are easy

25 Practice Questions

Question 1 Easy

Question 2 Easy

Question 3 Easy

Question 4 Easy

Question 5 Easy

Question 6 Easy

Question 7 Easy

Question 8 Easy

Question 9 Easy

Question 10 Easy

Question 11 Medium

Question 12 Medium

Question 13 Medium

Question 14 Medium

Question 15 Medium

Question 16 Medium

Question 17 Medium

Question 18 Medium

Question 19 Medium

Question 20 Medium

Question 21 Hard

Question 22 Hard

Question 23 Hard

Question 24 Hard

Question 25 Hard

Answer key summary

About SOMATH & Kid Einsteins

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